Cat coughing/sneezing keeps coming back despite treatment?

I have an eight-year-old male cat. He has always been very healthy, but this year he’s had some respiratory problems. Here’s the history:

• In early March he started sneezing and coughing often.
• On 23 March, seen by vet, who gave Convenia injection and Metacam drops. He was fine for about 2.5 months, then started sneezing and coughing again.
• On 13 June, back to vet (different vet, same practice), gave Convenia + Metacam again. Same again, fine for several months, then started sneezing and coughing again.
• On 15 September, back to vet (third vet, same practice), who gave him a Colvasone injection and said to come back for a chest X-ray.
• On 23 September, had a chest X-ray, and given another Colvasone injection.

The vet said he thought it could be asthma, and the cat should be okay now for about a month but to come back if coughing recurred after that and he could be given a steroid mask. The receptionist said that the X-ray showed ‘patterns of feline asthma’ (we think – wasn’t totally clear).

It’s now only 28 September and he’s started sneezing again, about 10-20 times per hour, and snuffling. He’s also lethargic and off his food, which is very uncharacteristic!

My questions 🙁

• Does this sound like asthma, given that it’s come back again so quickly after a steroid injection?
• Is it normal to have two steroid injections eight days apart?
• If not asthma, what else could it be? (The vet said he thought it was unlikely to be a physical blockage, given that it responded to earlier treatment.)